Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 421 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 33 + 75\cdot 421 + 177\cdot 421^{2} + 328\cdot 421^{3} + 370\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 104 + 32\cdot 421 + 269\cdot 421^{2} + 397\cdot 421^{3} + 51\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 114 + 168\cdot 421 + 63\cdot 421^{2} + 401\cdot 421^{3} + 413\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 155 + 390\cdot 421 + 231\cdot 421^{2} + 85\cdot 421^{3} + 214\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 201 + 238\cdot 421 + 390\cdot 421^{2} + 12\cdot 421^{3} + 327\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 336 + 248\cdot 421 + 167\cdot 421^{2} + 220\cdot 421^{3} + 11\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 370 + 64\cdot 421 + 228\cdot 421^{2} + 316\cdot 421^{3} + 407\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 373 + 44\cdot 421 + 156\cdot 421^{2} + 342\cdot 421^{3} + 307\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,7,6)(3,4,8,5)$ |
| $(1,4,2,3,7,5,6,8)$ |
| $(1,7)(2,6)$ |
| $(1,7)(2,6)(3,8)(4,5)$ |
| $(1,8)(2,5)(3,7)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,7)(2,6)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,5)(3,7)(4,6)$ | $0$ |
| $4$ | $2$ | $(2,6)(3,4)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,2,7,6)(3,5,8,4)$ | $0$ |
| $2$ | $4$ | $(1,2,7,6)(3,4,8,5)$ | $0$ |
| $4$ | $4$ | $(1,8,7,3)(2,5,6,4)$ | $0$ |
| $4$ | $8$ | $(1,4,2,3,7,5,6,8)$ | $0$ |
| $4$ | $8$ | $(1,4,6,8,7,5,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
